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The Baer–Kaplansky Theorem for all abelian groups and modules

Simion Breaz Orcid Logo, Tomasz Brzezinski Orcid Logo

Bulletin of Mathematical Sciences, Volume: 12, Issue: 01, Pages: 1 - 12

Swansea University Author: Tomasz Brzezinski Orcid Logo

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DOI (Published version): 10.1142/s1664360721500053

Abstract

It is shown that the Baer-Kaplansky theorem can be extended to all abelian groups provided that the rings of endomorphisms of groups are replaced by trusses of endomorphisms of corresponding heaps. That is, every abelian group is determined up to isomorphism by its endomorphism truss and every isomo...

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Published in: Bulletin of Mathematical Sciences
ISSN: 1664-3607 1664-3615
Published: World Scientific Pub Co Pte Ltd 2021
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URI: https://cronfa.swan.ac.uk/Record/cronfa56478
first_indexed 2021-03-22T09:23:35Z
last_indexed 2022-04-09T03:19:48Z
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spelling 2022-04-08T12:22:33.9913900 v2 56478 2021-03-22 The Baer–Kaplansky Theorem for all abelian groups and modules 30466d840b59627325596fbbb2c82754 0000-0001-6270-3439 Tomasz Brzezinski Tomasz Brzezinski true false 2021-03-22 MACS It is shown that the Baer-Kaplansky theorem can be extended to all abelian groups provided that the rings of endomorphisms of groups are replaced by trusses of endomorphisms of corresponding heaps. That is, every abelian group is determined up to isomorphism by its endomorphism truss and every isomorphism between two endomorphism trusses associated to some abelian groups $G$ and $H$ is induced by an isomorphism between $G$ and $H$ and an element from $H$. This correspondence is then extended to all modules over a ring by considering heaps of modules. It is proved that the truss of endomorphisms of a heap associated to a module $M$ determines $M$ as a module over its endomorphism ring. Journal Article Bulletin of Mathematical Sciences 12 01 1 12 World Scientific Pub Co Pte Ltd 1664-3607 1664-3615 Abelian group; heap; endomorphism truss 13 4 2021 2021-04-13 10.1142/s1664360721500053 COLLEGE NANME Mathematics and Computer Science School COLLEGE CODE MACS Swansea University 2022-04-08T12:22:33.9913900 2021-03-22T09:19:01.1258882 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Simion Breaz 0000-0002-8506-4526 1 Tomasz Brzezinski 0000-0001-6270-3439 2 56478__19683__6b14e588c7d94a518a411f7685e36a36.pdf 56478.pdf 2021-04-19T13:27:47.1372582 Output 289169 application/pdf Version of Record true © The Author(s). It is distributed under the terms of the Creative Commons Attribution 4.0 (CC BY) License true eng https://creativecommons.org/licenses/by/4.0
title The Baer–Kaplansky Theorem for all abelian groups and modules
spellingShingle The Baer–Kaplansky Theorem for all abelian groups and modules
Tomasz Brzezinski
title_short The Baer–Kaplansky Theorem for all abelian groups and modules
title_full The Baer–Kaplansky Theorem for all abelian groups and modules
title_fullStr The Baer–Kaplansky Theorem for all abelian groups and modules
title_full_unstemmed The Baer–Kaplansky Theorem for all abelian groups and modules
title_sort The Baer–Kaplansky Theorem for all abelian groups and modules
author_id_str_mv 30466d840b59627325596fbbb2c82754
author_id_fullname_str_mv 30466d840b59627325596fbbb2c82754_***_Tomasz Brzezinski
author Tomasz Brzezinski
author2 Simion Breaz
Tomasz Brzezinski
format Journal article
container_title Bulletin of Mathematical Sciences
container_volume 12
container_issue 01
container_start_page 1
publishDate 2021
institution Swansea University
issn 1664-3607
1664-3615
doi_str_mv 10.1142/s1664360721500053
publisher World Scientific Pub Co Pte Ltd
college_str Faculty of Science and Engineering
hierarchytype
hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
document_store_str 1
active_str 0
description It is shown that the Baer-Kaplansky theorem can be extended to all abelian groups provided that the rings of endomorphisms of groups are replaced by trusses of endomorphisms of corresponding heaps. That is, every abelian group is determined up to isomorphism by its endomorphism truss and every isomorphism between two endomorphism trusses associated to some abelian groups $G$ and $H$ is induced by an isomorphism between $G$ and $H$ and an element from $H$. This correspondence is then extended to all modules over a ring by considering heaps of modules. It is proved that the truss of endomorphisms of a heap associated to a module $M$ determines $M$ as a module over its endomorphism ring.
published_date 2021-04-13T08:00:30Z
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score 11.364387

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